In this previous question of mine I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture which states that $ \forall\varepsilon>0,\forall x>2,\max \{r_{0}(n),n\leq x\}\ll_{\varepsilon}x^{\varepsilon}$, from which I derive an asymptotics for $ \lim\inf_{n\to\infty} p_{n+k}-p_{n} $ that is a consequence of Hardy-Littlewood k-tuple conjecture.
I would like to know whether the conjunction of Goldbach and NFPR conjectures is actually equivalent to Hardy-Littlewood's or if a more precise, $ \varepsilon $ free upper bound for $ r_{0}(n) $ is needed : for example do we need a small enough $ \alpha $ such that $ r_{0}(n)<C\log^{\alpha}n $ for all sufficiently large $ n $ and some absolute positive constant $ C $ and if so would $ \alpha=2 $ do the job ?
